Embedding of continuous fields of C* -algebras in the Cuntz algebra 𝒪2

Abstract

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra O 2 . We further prove that if A is a simple separable unital nuclear C*-algebra, then O 2 ⊗ A ∼ = O 2 , and if, in addition, A is purely infinite, then O∞ ⊗ A ∼ = A.

The one-cocycle property for shifts

“…First we note that O n is a purely infinite, simple, nuclear C * -algebra [5] and such a class is now well studied [12,13]. The equivalence of (1) and (2) follows from [14,22].…”

Non-commutative entropy computations for continuous fields and cross-products

“…In [4], Blanchard showed that exact continuous fields over a compact space X have C(X)-embeddings but in [10] for X = [0, 1] the authors proved the existence of lipschitz embeddings when an intrinsically defined metric function is itself lipschitz (see theorem 2.10 p.83). It is the case for example of the continuous field of the non-commutative tori (reproving a theorem of Haagerup-Rordam, see [9]).…”

“…Then a subtlety arises as it is not true that the whole algebra is exact whenever all fiber algebras are (even for continuous fields see [4]). Therefore we turned our attention to lipschitz continuous fields introduced by Kirchberg and Phillips [10] because, for such continuous fields, explicit matrix factorizations can be realized via the knowledge of factorizations of the fibers. We then found an upper bound for the entropy of an automorphism of such fields that has an extra term which incorporates geometric data (dimension of the base space, lipschitz exponent of the field) and a symbolic dynamics entropy term.…”

Lifting of an Asymptotically Inner Flow for a Separable C*-Algebra

“…We know, for single automorphisms, that the class of asymptotically inner automorphisms is smaller than the class of approximately inner automorphisms in general (see [5,6]). …”

“…There was another type of flows I reported on in my talk, namely, the Rohlin flows, which are far from the asymptotically inner flows but could be more manageable by its strong property of cocycle vanishing (at least when the C * -algebra is a Kirchberg algebra [5,6]). I will not discuss it here (for interested readers see [9,10,11]).…”